Abstract:
In this talk, we review the heat kernel approach to the Atiyah-Singer index theorem for Dirac operators on closed manifolds, as well as the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. We also discuss the odd dimensional counterparts of the above results. In particular, we describe a joint result with Xianzhe Dai on an index theorem for Toeplitz operators on odd dimensional manifolds with boundary.

Abstract:
We show that the R/Z part of the analytically defined eta invariant of Atiyah-Patodi-Singer for a Dirac operator on an odd dimensional closed spin manifold can be expressed purely geometrically through a stable Chern-Simons current on a higher dimensional sphere. As a preliminary application, we discuss the relation with the Atiyah-Patodi-Singer R/Z index theorem for unitary flat vector bundles, and prove an R refinement in the case where the Dirac operator is replaced by the Signature operator. We also extend the above discussion to the case of eta invariants associated to Hermitian vector bundles with non-unitary connection, which are constructed by using a trick due to Lott.

Abstract:
We establish an S^1-equivariant index theorem for Dirac operators on Z/k-manifolds. As an application, we generalize the Atiyah-Hirzebruch vanishing theorem for S^1-actions on closed spin manifolds to the case of Z/k-manifolds.

Abstract:
We construct Dirac operators on foliations by applying the Bismut-Lebeau analytic localization technique to the Connes fibration over a foliation. The Laplacian of the resulting Dirac operators has better lower bound than that obtained by using the usual adiabatic limit arguments on the original foliation. As a consequence, we prove an extension of the Lichnerowicz-Hitchin vanishing theorem to the case of foliations.

Abstract:
We prove a Lichnerowicz type vanishing theorem for non-compact spin manifolds admiting proper cocompact actions. This extends a previous result of Ziran Liu who proves it for the case where the acting group is unimodular.

Abstract:
We prove the following generalization of the Lichnerowicz-Hitchin vanishing theorem to the case of foliations: let $M$ be a closed spin manfold, let $F$ be an integrable subbundle of the tangent bundle $TM$ such that $F$ carries a metric of positive leafwise scalar curvature, then the canonical $KO$-characteristic number $\hat{\mathcal A}(M)$ vanishes. Our proof applies to give a geometric proof of the Connes vanishing theorem, which states that in the case of $F$ being spin instead of $TM$ being spin, one has $\hat{A}(M)=0$.

Abstract:
We establish a mod 2 index theorem for real vector bundles over 8k+2 dimensional compact pin$^-$ manifolds. The analytic index is the reduced $\eta$ invariant of (twisted) Dirac operators and the topological index is defined through $KO$-theory. Our main result extends the mod 2 index theorem of Atiyan and Singer to non-orientable manifolds.

Abstract:
We show that the holomorphic Morse inequalities proved by Tian and the author [TZ1, 2] are in effect equalities by refining the analytic arguments in [TZ1, 2].

Abstract:
We give a brief survey on aspects of the local index theory as developed from the mathematical works of V. K. Patodi. It is dedicated to the 70th anniversary of Patodi.